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Popolare Algebra >

espandere (2x^2+x)^{12}

Soluzione

espandere

(2 x^{2} + x)^{12}

Soluzione

4096 x^{24} + 24576 x^{23} + 67584 x^{22} + 112640 x^{21} + 126720 x^{20} + 101376 x^{19} + 59136 x^{18} + 25344 x^{17} + 7920 x^{16} + 1760 x^{15} + 264 x^{14} + 24 x^{13} + x^{12}

Fasi della soluzione

(2 x^{2} + x)^{12}

Applicare il teorema binomiale:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 2 x^{2}, b = x

= i = 0 \sum 12 (i 12) (2 x^{2})^{(12 - i)} x^{i}

Espandere la sommatoria

= \frac{12 !}{0 ! ( 12 - 0 ) !} (2 x^{2})^{12} x^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} (2 x^{2})^{11} x^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} (2 x^{2})^{10} x^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} (2 x^{2})^{9} x^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} (2 x^{2})^{8} x^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} (2 x^{2})^{7} x^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} (2 x^{2})^{6} x^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} (2 x^{2})^{5} x^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} (2 x^{2})^{4} x^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} (2 x^{2})^{3} x^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} (2 x^{2})^{2} x^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} (2 x^{2})^{1} x^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} (2 x^{2})^{0} x^{12}

Semplificare

\frac{12 !}{0 ! ( 12 - 0 ) !} (2 x^{2})^{12} x^{0} : 4096 x^{24}

Semplificare

\frac{12 !}{1 ! ( 12 - 1 ) !} (2 x^{2})^{11} x^{1} : 24576 x^{23}

Semplificare

\frac{12 !}{2 ! ( 12 - 2 ) !} (2 x^{2})^{10} x^{2} : 67584 x^{22}

Semplificare

\frac{12 !}{3 ! ( 12 - 3 ) !} (2 x^{2})^{9} x^{3} : 112640 x^{21}

Semplificare

\frac{12 !}{4 ! ( 12 - 4 ) !} (2 x^{2})^{8} x^{4} : 126720 x^{20}

Semplificare

\frac{12 !}{5 ! ( 12 - 5 ) !} (2 x^{2})^{7} x^{5} : 101376 x^{19}

Semplificare

\frac{12 !}{6 ! ( 12 - 6 ) !} (2 x^{2})^{6} x^{6} : 59136 x^{18}

Semplificare

\frac{12 !}{7 ! ( 12 - 7 ) !} (2 x^{2})^{5} x^{7} : 25344 x^{17}

Semplificare

\frac{12 !}{8 ! ( 12 - 8 ) !} (2 x^{2})^{4} x^{8} : 7920 x^{16}

Semplificare

\frac{12 !}{9 ! ( 12 - 9 ) !} (2 x^{2})^{3} x^{9} : 1760 x^{15}

Semplificare

\frac{12 !}{10 ! ( 12 - 10 ) !} (2 x^{2})^{2} x^{10} : 264 x^{14}

Semplificare

\frac{12 !}{11 ! ( 12 - 11 ) !} (2 x^{2})^{1} x^{11} : 24 x^{13}

Semplificare

\frac{12 !}{12 ! ( 12 - 12 ) !} (2 x^{2})^{0} x^{12} : x^{12}

= 4096 x^{24} + 24576 x^{23} + 67584 x^{22} + 112640 x^{21} + 126720 x^{20} + 101376 x^{19} + 59136 x^{18} + 25344 x^{17} + 7920 x^{16} + 1760 x^{15} + 264 x^{14} + 24 x^{13} + x^{12}

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